Theorem opeq 4620

Description: Any class is equal to an ordered pair. (Contributed by Scott Fenton, 8-Apr-2021.)

Assertion

Ref	Expression
opeq	|- A = <. Proj1 A, Proj2 A>.

Proof of Theorem opeq

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-op 4567	. 2 |- <. Proj1 A, Proj2 A>. = ({x | E.y e. Proj1 Ax = Phi y} u. {x | E.y e. Proj2 Ax = ( Phi y u. {0c})})
2		df-proj1 4568	. . . . . . 7 |- Proj1 A = {z | Phi z e. A}
3	2	rexeqi 2813	. . . . . 6 |- (E.y e. Proj1 Ax = Phi y <-> E.y e. {z | Phi z e. A}x = Phi y)
4		phieq 4571	. . . . . . . 8 |- (z = y -> Phi z = Phi y)
5	4	eleq1d 2419	. . . . . . 7 |- (z = y -> ( Phi z e. A <-> Phi y e. A))
6	5	rexab 3000	. . . . . 6 |- (E.y e. {z | Phi z e. A}x = Phi y <-> E.y( Phi y e. A /\ x = Phi y))
7		ancom 437	. . . . . . . . 9 |- (( Phi y e. A /\ x = Phi y) <-> (x = Phi y /\ Phi y e. A))
8		eleq1 2413	. . . . . . . . . 10 |- (x = Phi y -> (x e. A <-> Phi y e. A))
9	8	pm5.32i 618	. . . . . . . . 9 |- ((x = Phi y /\ x e. A) <-> (x = Phi y /\ Phi y e. A))
10	7, 9	bitr4i 243	. . . . . . . 8 |- (( Phi y e. A /\ x = Phi y) <-> (x = Phi y /\ x e. A))
11	10	exbii 1582	. . . . . . 7 |- (E.y( Phi y e. A /\ x = Phi y) <-> E.y(x = Phi y /\ x e. A))
12		19.41v 1901	. . . . . . 7 |- (E.y(x = Phi y /\ x e. A) <-> (E.y x = Phi y /\ x e. A))
13		ancom 437	. . . . . . 7 |- ((E.y x = Phi y /\ x e. A) <-> (x e. A /\ E.y x = Phi y))
14	11, 12, 13	3bitri 262	. . . . . 6 |- (E.y( Phi y e. A /\ x = Phi y) <-> (x e. A /\ E.y x = Phi y))
15	3, 6, 14	3bitri 262	. . . . 5 |- (E.y e. Proj1 Ax = Phi y <-> (x e. A /\ E.y x = Phi y))
16	15	abbii 2466	. . . 4 |- {x | E.y e. Proj1 Ax = Phi y} = {x | (x e. A /\ E.y x = Phi y)}
17		df-rab 2624	. . . 4 |- {x e. A | E.y x = Phi y} = {x | (x e. A /\ E.y x = Phi y)}
18	16, 17	eqtr4i 2376	. . 3 |- {x | E.y e. Proj1 Ax = Phi y} = {x e. A | E.y x = Phi y}
19		df-proj2 4569	. . . . . . 7 |- Proj2 A = {z | ( Phi z u. {0c}) e. A}
20	19	rexeqi 2813	. . . . . 6 |- (E.y e. Proj2 Ax = ( Phi y u. {0c}) <-> E.y e. {z | ( Phi z u. {0c}) e. A}x = ( Phi y u. {0c}))
21	4	uneq1d 3418	. . . . . . . 8 |- (z = y -> ( Phi z u. {0c}) = ( Phi y u. {0c}))
22	21	eleq1d 2419	. . . . . . 7 |- (z = y -> (( Phi z u. {0c}) e. A <-> ( Phi y u. {0c}) e. A))
23	22	rexab 3000	. . . . . 6 |- (E.y e. {z | ( Phi z u. {0c}) e. A}x = ( Phi y u. {0c}) <-> E.y(( Phi y u. {0c}) e. A /\ x = ( Phi y u. {0c})))
24		ancom 437	. . . . . . . . 9 |- ((( Phi y u. {0c}) e. A /\ x = ( Phi y u. {0c})) <-> (x = ( Phi y u. {0c}) /\ ( Phi y u. {0c}) e. A))
25		eleq1 2413	. . . . . . . . . 10 |- (x = ( Phi y u. {0c}) -> (x e. A <-> ( Phi y u. {0c}) e. A))
26	25	pm5.32i 618	. . . . . . . . 9 |- ((x = ( Phi y u. {0c}) /\ x e. A) <-> (x = ( Phi y u. {0c}) /\ ( Phi y u. {0c}) e. A))
27	24, 26	bitr4i 243	. . . . . . . 8 |- ((( Phi y u. {0c}) e. A /\ x = ( Phi y u. {0c})) <-> (x = ( Phi y u. {0c}) /\ x e. A))
28	27	exbii 1582	. . . . . . 7 |- (E.y(( Phi y u. {0c}) e. A /\ x = ( Phi y u. {0c})) <-> E.y(x = ( Phi y u. {0c}) /\ x e. A))
29		19.41v 1901	. . . . . . 7 |- (E.y(x = ( Phi y u. {0c}) /\ x e. A) <-> (E.y x = ( Phi y u. {0c}) /\ x e. A))
30		ancom 437	. . . . . . 7 |- ((E.y x = ( Phi y u. {0c}) /\ x e. A) <-> (x e. A /\ E.y x = ( Phi y u. {0c})))
31	28, 29, 30	3bitri 262	. . . . . 6 |- (E.y(( Phi y u. {0c}) e. A /\ x = ( Phi y u. {0c})) <-> (x e. A /\ E.y x = ( Phi y u. {0c})))
32	20, 23, 31	3bitri 262	. . . . 5 |- (E.y e. Proj2 Ax = ( Phi y u. {0c}) <-> (x e. A /\ E.y x = ( Phi y u. {0c})))
33	32	abbii 2466	. . . 4 |- {x | E.y e. Proj2 Ax = ( Phi y u. {0c})} = {x | (x e. A /\ E.y x = ( Phi y u. {0c}))}
34		df-rab 2624	. . . 4 |- {x e. A | E.y x = ( Phi y u. {0c})} = {x | (x e. A /\ E.y x = ( Phi y u. {0c}))}
35	33, 34	eqtr4i 2376	. . 3 |- {x | E.y e. Proj2 Ax = ( Phi y u. {0c})} = {x e. A | E.y x = ( Phi y u. {0c})}
36	18, 35	uneq12i 3417	. 2 |- ({x | E.y e. Proj1 Ax = Phi y} u. {x | E.y e. Proj2 Ax = ( Phi y u. {0c})}) = ({x e. A | E.y x = Phi y} u. {x e. A | E.y x = ( Phi y u. {0c})})
37		unrab 3527	. . 3 |- ({x e. A | E.y x = Phi y} u. {x e. A | E.y x = ( Phi y u. {0c})}) = {x e. A | (E.y x = Phi y \/ E.y x = ( Phi y u. {0c}))}
38		rabid2 2789	. . . 4 |- (A = {x e. A | (E.y x = Phi y \/ E.y x = ( Phi y u. {0c}))} <-> A.x e. A (E.y x = Phi y \/ E.y x = ( Phi y u. {0c})))
39		vex 2863	. . . . . . 7 |- x e. _V
40	39	phiall 4619	. . . . . 6 |- E.y(x = Phi y \/ x = ( Phi y u. {0c}))
41		19.43 1605	. . . . . 6 |- (E.y(x = Phi y \/ x = ( Phi y u. {0c})) <-> (E.y x = Phi y \/ E.y x = ( Phi y u. {0c})))
42	40, 41	mpbi 199	. . . . 5 |- (E.y x = Phi y \/ E.y x = ( Phi y u. {0c}))
43	42	a1i 10	. . . 4 |- (x e. A -> (E.y x = Phi y \/ E.y x = ( Phi y u. {0c})))
44	38, 43	mprgbir 2685	. . 3 |- A = {x e. A | (E.y x = Phi y \/ E.y x = ( Phi y u. {0c}))}
45	37, 44	eqtr4i 2376	. 2 |- ({x e. A | E.y x = Phi y} u. {x e. A | E.y x = ( Phi y u. {0c})}) = A
46	1, 36, 45	3eqtrri 2378	1 |- A = <. Proj1 A, Proj2 A>.

Syntax hints:   \/ wo 357   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2616  {crab 2619   u. cun 3208  {csn 3738  0_cc0c 4375  <.cop 4562   Phi cphi 4563   Proj1 cproj1 4564   Proj2 cproj2 4565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-0c 4378  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569

This theorem is referenced by:  opeqexb  4621  xpvv  4844  ssrel  4845  proj1eldm  4928  co01  5094  1stfo  5506  2ndfo  5507  swapf1o  5512  otsnelsi3  5806  xpassenlem  6057  xpassen  6058  nncdiv3lem1  6276  dmfrec  6317  fnfreclem2  6319